On Integral Lebesgue Constants of Local Splines with Uniform Knots

Resumo

We study the stability properties of generalized local splines of the form

\(S(x) = S(f,x) = \sum\limits_{j \in \mathbb{Z}} {{y_j}{B_\varphi }\left( {x + \frac{{3h}}{2} - jh} \right),}\;\;\; x \in \mathbb{R},\)

where ϕ ∈ C¹[−h, h] for h > 0, ϕ(0) = ϕ′(0) = 0, ϕ(−x) = ϕ(x)for x ∈ [0; h], ϕ(x) is nondecreasing on [0; h], f is an arbitrary function from ℝ to ℝ, y_j = f(jh) for j ∈ ℤ, and

\({B_\varphi }(x) = m(h)\left\{ {\begin{array}{*{20}{c}}
{\varphi (x),}&{x \in [0;h],} \\
{2\varphi (h) - \varphi (x - h) - \varphi (2h - x),}&{x \in [h;2h],} \\
{\varphi (3h - x),}&{x \in [2h;3h],} \\
{0,}&{x \notin [0;3h],}
\end{array}} \right.\)

with m(h) > 0. Such splines were constructed by the author earlier. In the present paper, we calculate the exact values of their integral Lebesgue constants (the norms of linear operators from l to L) on the axis ℝ and on any segment of the axis for a certain choice of the boundary conditions and the normalizing factor m(h) of the spline S.